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Name Student ID # Math 1320-004 April 24, 2014 FINAL EXAM Instructions: Show all of your work for full credit. Problem 1. 2. 3. 4. 5. Points 20 20 20 20 20 TOTAL Score (20 points) 1. Suppose r(t) = ht + 2, 1, t2 i gives the position of an object at time t. Find the unit tangent vector to the path, and the tangential component of acceleration, at time t = 1. (20 points) 2. An angry bird is launched from a slingshot on the (flat) surface of a planet, at an angle of 60◦ from horizontal, and leaves the slingshot at 61 m. Acceleration due to gravity is 2 m/s2 . How 1 m/s at a height of 16 far does the bird travel in the horizontal direction? 1 (20 points) 3. Suppose v(x, y) = x2 y + ey , where x(r, s) and y(r, s) are unknown. If x(1, 2) = 1, xr (1, 2) = −3, xs (1, 2) = 5, y(1, 2) = −4, yr (1, 2) = −1, ∂v (1, 2). and ys (1, 2) = 2, find ∂s (20 points) 4. Find and classify (using the Second Derivatives Test) local maxima, local minima, or saddle points of the function f (x, y) = x4 + y 4 − 4xy. 2 (20 points) 5. Find the absolute minimum and absolute maximum of f (x, y) = x2 ye−x−y over the square with vertices (0, 0), (3, 0), (0, 3), and (3, 3). 3